Good chance one of them is me, i have never ending storage problems
2 solutions, pick the one that applies:
1) Stop hoarding so much stuff. most people are holding onto way more items than they need to. Don't be scared to throw things away that you're not using. Personally I only hold Max Potions and Max Revives any of the lower types are insta-discarded unless they can be used immediately. Used to discard anything above 100 revives/50 potions but I scaled back to 80/40 and I'm considering dropping even lower as it's rare for me to get more than 10-20 into those reserves before they're topped up again. Similar with balls. At the moment, circumstances dictate that I'm almost exclusively using the Gotcha for spins/catches. As a result my stocks of Blue/Yellow balls are getting out of hand and I'm having to start discarding blues to make more room for reds.
2) Buy more storage. Trying to play with little more than the standard amount of storage is painful. Buying more storage is one of he best uses for your free coins as its a one of purchase for a permanent benefit.
Finally, even if your storage is maxed try to open gifts anyway, even if it means throwing away the stuff you don't need. Levelling up friendships is the key to big XP gains, you'll get more gifts anyway and it's another way to try and tune the split of your bag contents to fit what you want to be carrying.
Obviously if you're the kind of wierdo who has no bag space because you're is holding onto 1000 rare candy and 300 raid passes then you already know all that and I have no sympathy.